1. Introduction

Synchronization phenomena in coupled semiconductor lasers have attracted lot of attention, motivated, on one hand, for the practical applications of synchronized optical chaos [1], and on the other hand, from a nonlinear science perspective, for the rich variety of complex dynamical behaviors that generically occurs in interacting nonlinear systems [2]. When two lasers are optically coupled, there is a delay time in their coupling due to the finite flight time from one laser to the other, and such delayed coupling can induce a rich variety of complex phenomena, including isochronous synchronization [3, 4, 5, 6], time leadership competition and intermittent time leadership exchange [7, 8].

A particular way of generating synchronized optical chaos is by cross-coupling two lasers, in a coupling scheme such that the intensity of one laser is used as an input to modulate another variable of the other laser, e.g., via opto-electronic coupling [9, 10]. This scheme can also be implemented all-optically, by polarization-rotated coupling [11, 12, 13, 14]: the light emitted by one laser passes through a polarization-rotating device (that rotates the light polarization by 90 degrees) and is then injected into the other laser, and vice-versa.

A modification of this setup includes a linear polarizer that selects only one linear polarization in the emitted output of one laser, that is rotated and then injected into the other laser, and vice-versa. With two edge-emitting lasers (EELs) mutually coupled with such polarization-rotated scheme, square-wave (SW) polarization switchings have been observed experimentally, with periodicity slightly larger than twice the coupling delay time [15].

Such regular all-optical square-wave switching could have potential applications in communications and information processing technologies. In [15] the regular polarization switching dynamics was interpreted in terms of a two-mode rate-equation model that predicts that this dynamics is actually a transient dynamics, that can be experimentally sustained by noise; however, further numerical investigations [16] have shown that, when self- and cross- gain saturation and a frequency detuning between the two polarizations are included in the model, the regular switching dynamics can be numerically stable in certain parameter regions.

The aim of this work is to investigate if, in vertical-cavity surface-emitting lasers (VCSELs), regular square-wave polarization switching can be, under appropriated conditions, a stable dynamical regime. VCSELs are nowadays widely employed in optical systems [17] and regular all-optical square-wave switching could have promising applications. Our study is motivated by the fact that VCSELs present a polarization behavior that is very different from that of EELs [18, 19].

We use as a framework the spin-flip model [20], extended as in [15] to account for delayed and polarization-rotated mutual coupling. We find that in this model square-wave polarization switching is a transient effect towards a stationary state, and the lasers’ dynamics on the stationary state depends on the various model parameters: the lasers can either emit a linearly polarized output on two orthogonal polarizations (referred to as x and y, and while one laser emits a x-polarized output, the other laser emits a y-polarized output), or they can both emit on the two polarizations. In addition, the polarization intensities that can be either constant in time in the two lasers, or time-dependent in the two lasers, or one laser can emit a cw intensity while the other, a time-dependent one.

This article is organized as follows: Section 2 briefly presents the spin-flip model, extended to account for mutual orthogonal coupling; Sec. 3 presents the numerical results. The influence of various parameters on the duration of the transient time, as well as on the lasers’ stationary dynamics is analyzed. Section 4 presents a summary of the results and the conclusions.

2. Model

2.1. Rate equations

The model equations are [15, 20]

Here i = 1 and i = 2 denote the two lasers, E_x and E_y are slowly-varying complex amplitudes, N and n are two carrier densities (N = N₊ + N₋, n = N₊ – N₋ with N₊ and N₋ being carrier populations with opposite spin), k is the field decay rate, γ_n is the carrier decay rate, γ_s is the spin-flip rate, α the linewidth enhancement factor, γ_a and γ_p are anisotropies representing dichroism and birefringence: for γ_a > 0 (γ_p > 0) the y polarization has a lower threshold (a higher frequency) than the x polarization. β_sp is the strength of spontaneous emission noise, ξ_x,y are uncorrelated Gaussian white noises and μ is the injection current parameter, normalized such that the threshold of the solitary lasers in the absence of anisotropies is at μ_th,s = 1.

The coupling parameters are the delay time, τ, and the coupling strength, η: when the x polarization of one laser is injected into the y polarization of the other laser (x → y), η_x = η and η_y = 0, and when y → x, η_x = 0 and η_y = η. Because of the interplay of birefringence (represented by the parameter γ_p) and the phase-amplitude coupling (represented by the α factor), these two coupling schemes are not symmetric, and the frequency difference between the two polarizations plays a role similar to that of the detuning between an injected and a master laser.

2.2. Steady state solutions

The model has two types of steady states: “pure modes” [15], in which the lasers emit orthogonal polarizations, and “mixed modes”, in which the intensities of the two polarizations are non zero in the two lasers.

There are two pure mode solutions:

These pure mode solutions can be calculated analytically, following a similar procedure as that discussed in Ref. [16] for a model for edge-emitting lasers. In the case of x → y coupling we obtain the equation:

The mixed states are symmetric and have I_1x = I_2x; I_1y = I_2y; however, their equations are too complicated to solve analytically. They can be found numerically for low coupling strengths, as will be discussed in the next section.

3. Results

To simulate the model equations, unless otherwise specifically stated, we use the following parameter values: k = 300 ns⁻¹, γ_n = 2 ns⁻¹, γ_s = 50 ns⁻¹, γ_a = 0.4 ns⁻¹, α = 3, β_sp = 10⁻⁵ ns⁻¹, and τ = 3 ns. The pump current parameter and the birefringence parameter are chosen such that the solitary lasers display polarization mono-stability (the stability region of the two linear polarizations is displayed Fig. 1). For a pump current parameter about twice the threshold, μ ≃ 2, the x polarization is stable if the birefringence is large enough (γ_p ≥ 50 rad/ns), while the y polarization is stable if the birefringence is low enough (γ_p ≤ 8 rad/ns). The simulations start with initial conditions are such that both lasers are off.

 

Fig. 1 Linear stability of the x and y polarized states of the solitary lasers in the parameter space (birefringence, injection current). In the red region only the x polarization is stable, in the blue region, only the y polarization is stable, in the white region, both polarizations are stable and in the green region, neither polarization is stable. The model parameters are: k = 300 ns⁻¹, γ_n = 2 ns⁻¹, γ_s = 50 ns⁻¹, γ_a = 0.4 ns⁻¹ and α = 3. The black circle and square indicate the parameters used for Figs. 3(a) and (b) respectively.

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Figure 2 displays the pure mode solution calculated analytically (black dots) together with the numerical solution (the extreme values of the oscillations of the x intensity in one laser in red, and the extreme values of the oscillations of the y intensity in the other laser in blue), as a function of the coupling strength. In Figs. 2(a), (b) the coupling is x → y while in Figs. 2(c), (d) the coupling is y → x. One can notice that in both cases, for strong coupling the pure mode is dominant, while for low coupling, the mixed state dominates. Analytically we find that, depending on the coupling strength, there are either one or three pure mode solutions. The stability analysis of these solutions can not be done analytically but numerically we find that the middle and lower solutions are unstable, while the upper solution is stable as soon as it exists [as in Fig. 2(b)] or becomes stable after a series of bifurcations [as in Fig. 2(c)].

 

Fig. 2 Bifurcation diagram for increasing coupling strength. The analytically calculated pure mode solution (black dots) is plotted together with the numerical solution (the extreme values of the oscillations of the x intensity of one laser, I_1x in red, and the extreme values of the oscillations of the y intensity of the other laser, I_2y in blue). In (a), (b) the coupling is x → y, in (c), (d) the coupling is y → x. The parameters are μ = 2, γ_p = 60 rad/ns (a), (b), γ_p = 4 rad/ns (c), (d), other parameters are as indicated in the text.

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Figure 3 displays the transient behavior towards a pure mode solution in which the lasers emit cw light with orthogonal polarizations (in both lasers, the intensity of the x-polarization in red and the intensity of the y-polarization in blue; the intensities were averaged to simulate the finite experimental detection bandwidth). During the transient there are square-wave polarization switchings, and it can be noticed that the lasers emit the same polarization and switch simultaneously to the orthogonal one. The sum of two consecutive switching intervals is slightly larger than 2τ. This dynamical switching is metastable and eventually evolves as shown in the right panels: the intervals that one laser emits one polarization grow, while the intervals that emits the orthogonal one decrease, until a steady-state is reached where the lasers emit polarization-orthogonal cw outputs. In this “pure-mode” one laser emits its natural polarization and acts as a “master laser” injecting polarization-rotated cw light into the other laser (referred to as “injected laser”). The injection is strong enough to turn off the natural polarization of the injected laser and turn on the orthogonal one. Since the lasers are coupled via their natural polarizations, the “master laser” does not receive light from the “injected laser” and therefore, their coupling is in fact unidirectional.

 

Fig. 3 Transient dynamics towards the “pure-mode” steady-state in which the lasers emit cw orthogonal polarizations. The intensities of the x and y polarizations of the two lasers are plotted vs. time (x red line; y blue line, the intensities were averaged to simulate the finite experimental detection bandwidth). Left panels: the coupling is x → y and the parameters are μ = 2.5, γ_p = 60 rad/ns, η = 60 ns⁻¹; right panels: y → x, μ = 1.7, γ_p = 4 rad/ns, η = 80 ns⁻¹, other parameters as indicated in the text. In both cases after a transient time laser 2 (bottom row) acts as “master laser”, emitting the natural polarization of the stand alone laser (x in the left panels and y in the right panels), while laser 1 (top row) is the “injected laser”, emitting the orthogonal polarization.

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For weaker coupling three qualitatively different behaviors are found, depending on the parameters:

For large enough values of the pump current parameter the x polarization of the solitary laser loses stability (see the green region in Fig. 1) and the dominant x polarization acquires a small component in the orthogonal y polarization. For even larger pump current the intensities I_x and I_y of the solitary laser both display regular oscillations. In this situation, when the lasers are mutually coupled, the “master” laser exhibits oscillations in both polarizations (the “solitary laser” solution) while the “injected” laser emits only the y polarization and shows the same oscillations as the master laser, as shown in the left panels of Fig. 4.

 

Fig. 4 Left panels: Dynamics of the coupled lasers with x → y coupling, μ = 3.5, other parameters as in the left panels of Fig. 3. After the transient laser 1 (top row) is the “master laser” displaying sustained oscillations of the two polarizations (“solitary laser solution”) and laser 2 (bottom row) is the “injected laser” emitting the y polarization with the same oscillation as the master. Right panels: Dynamics of the coupled lasers with weaker coupling strength and y → x coupling. After the transient laser 2 (bottom row) is the “injected laser” displaying sustained oscillations of the x polarization and laser 1 (top row) is the “master laser” emitting the y polarization with cw output (μ = 2, γ_p = 4 rad/ns, η = 50 ns⁻¹).

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While the analogy with orthogonally polarized optical injection is useful, one should however keep in mind that, besides the fact that there is mutual coupling, there is also a time delay in the coupling and thus, there is multistability in the form of coexisting solutions (as is typical in time-delayed systems). For example, there are parameter regions where both, polarization coexistence and single-polarization emission can be observed, depending on the stochastic trajectory.

We studied the influence of various model parameters on the statistics of the transient time. For each parameter value we simulated 30 stochastic trajectories with a maximum integration time of 10 μs (i.e., this is the longest transient time that we could compute). We limited the study to parameter regions where, for the solitary lasers, only one polarization is stable (regions red and blue in Fig. 1). We found that the transient time is a stochastic quantity that can display large deviations from its mean value. As shown in Fig. 5(a), the average duration of the transient time is not affected by the noise strength, if this is not too strong. The injection current parameter also does not seem to significantly affect the duration of the transient, Fig. 5(b). The average transient increases with the delay time, Fig. 6(a), and with the coupling strength, Fig. 6(b). The linear anisotropy parameter, γ_a, and the spin-flip rate, γ_s, do not significantly affect the duration of the transient, as shown in Figs. 7 (a), (b).

 

Fig. 5 Transient time vs. (a) the noise strength, β_sp, and (b) the injection current parameter, μ. For each parameter value we simulated 30 stochastic trajectories, with an integration time of 10 μs (i.e., this is the longest transient time that we can compute). The dashed line indicates the average value. In (a) μ = 2.5, in (b) β_sp = 10⁻⁵ ns⁻¹, other model parameters are k = 300 ns⁻¹, γ_n = 2 ns⁻¹, γ_s = 50 ns⁻¹, γ_a = 0.4 ns⁻¹, γ_p = 60 rad ns⁻¹, α = 3, η = 50 ns⁻¹ and τ = 3 ns.

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Fig. 6 As in Fig. 5 but plot of the transient time vs. (a) the delay time, τ, and (b) the coupling strength, η. In (a) η = 50 ns⁻¹, in (b) τ = 3 ns, other model parameters are k = 300 ns⁻¹, γ_n = 2 ns⁻¹, γ_s = 50 ns⁻¹, γ_a = 0.4 ns⁻¹, γ_p = 60 rad ns⁻¹, α = 3, μ = 2.5 and β_sp = 10⁻⁵ ns⁻¹.

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Fig. 7 As in Fig. 5 but plot of the transient time vs. (a) the anisotropy parameter, γ_a, and (b) the spin-flip rate, γ_s. In (a) γ_s = 50 ns⁻¹, in (b) γ_a = 0.4 ns⁻¹, other model parameters are k = 300 ns⁻¹, γ_n = 2 ns⁻¹, γ_p = 60 rad ns⁻¹, α = 3, μ = 2.5, β_sp = 10⁻⁵ ns⁻¹, η = 50 ns⁻¹ and τ = 3 ns.

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The results presented in Figs. 5, 6, and 7 are for x → y injection; for y → x injection we found qualitatively similar results, except on the influence of the birefringence parameter, γ_p, that is displayed in Fig. 8. For y → x injection there is no significant influence while for x → y there are three distinct regions: for γ_p ≤ 55 rad/ns (but larger than ≈50 rad/ns, such that for the solitary lasers only the x polarization is stable), the duration of the transient decreases with γ_p. For intermediate values of γ_p (55 rad/ns ≤ γ_p ≤ 65 rad/ns) the “pure-mode” solution is stable and the transient time decreases abruptly. For larger γ_p the transient time diverges.

 

Fig. 8 As in Fig. 5 but plot of the transient time vs. the birefringence parameter, γ_p, for (a) x → y injection and (b) y → x injection. In (a) γ_p = 60 rad ns⁻¹, in (b) γ_p = 6 rad ns⁻¹, other model parameters are k = 300 ns⁻¹, γ_n = 2 ns⁻¹, γ_s = 50 ns⁻¹, γ_a = 0.4 ns⁻¹, α = 3, μ = 2.5, β_sp = 10⁻⁵ ns⁻¹, η = 50 ns⁻¹ and τ = 3 ns.

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While we can not exclude the existence of parameter regions such that square-wave polarization switchings are not a transient behavior, in all the regions explored, such that the solitary lasers display polarization mono-stability (either in the x low-frequency mode, or in the y high-frequency mode), square-wave switchings were found to be a metastable dynamics, towards one of the various possible stationary states described above.

4. Discussion and conclusion

To conclude, we studied numerically the dynamics of two VCSELs with time-delayed coupling via the mutual injection of the natural lasing mode into the orthogonal one. We used the spin-flip model to simulate the VCSEL dynamics and considered parameters such that the solitary lasers displayed polarization mono-stability: they emitted the low-frequency, x-polarization, for high birefringence, or the high-frequency, y-polarization, for low birefringence.

We considered two types of polarization-rotated injection: from the x to the y polarization (x → y) and vice-versa (y → x). Because of the phase-amplitude coupling, represented by the α factor, these two coupling schemes are not symmetric; however, for strong enough coupling, in both cases we found a transient dynamics towards a stationary state characterized by square-wave-like polarization switchings of periodicity about twice the coupling delay time.

In the stationary state, depending on the parameters, either: (i) the two lasers emit on orthogonal linear polarizations; or (ii) the two lasers emit on both linear polarizations. When they emit orthogonal polarizations, the coupling is in fact uni-directional and one laser acts as a “solitary laser” emitting its “natural” lasing polarization, while the other laser acts as an “injected laser” emitting the orthogonal polarization. The “injected laser” can either emit a cw intensity (the so-called “pure-mode” solution), or can emit a time-dependent intensity displaying periodic or chaotic oscillations. When both lasers emit both polarizations, the lasers are effectively mutually coupled and they can either emit cw intensities (the so-called “mix-mode” solution) at low coupling strengths, or periodic or chaotic oscillations at stronger coupling strengths.

The duration of the transient dynamics was characterized in terms of various model parameters. The coupling parameters (the coupling strength and the delay time) significantly increase the average duration of the transient, while the noise strength and the pump current leave it nearly unaffected. The dichroism parameter and the spin-flip rate also do not affect the average transient time, while the birefringence can have a large impact, playing a role similar to that of the frequency detuning in a master-slave configuration.

While several of the dynamical behaviors found in coupled VCSELs are similar to those found in coupled edge-emitting lasers [16], we found also some important differences, due to the different polarization dynamics of standalone EELs and VCSELs: 1) in the VCSEL model no parameter region was found where polarization square-wave switching is a stable dynamics (in the EEL model there are narrow parameter regions where regular square-wave switching is stable [16]); 2) the asymmetry of the x → y and y → x coupling schemes results in different dynamical regimes at intermediate coupling strengths (this difference was not studied in [16]); 3) in the spin-flip VCSEL model there are parameter regions (at high pump currents) where, for the solitary laser, both polarizations are unstable, and in these regions we found that, for the coupled VCSELs, the “master laser” displays time-dependent oscillations in both polarizations which are transmitted to the “injected laser”, that displays the same oscillations but only in the orthogonal, naturally suppressed polarization.

Since the experimental observations reported in Refs. [15, 16] were done with coupled edge-emitting lasers, and we are not aware of any published experiments with VCSELs, we hope that our results will stimulate new experiments that will allow to test the model predictions.